CURRENT RESEARCH IN SOCIAL PSYCHOLOGY Volume 1, Number 5 Submitted March 5, 1996 Resubmitted March 29, 1996 Accepted April 3, 1996 Publication Date: April 26, 1996 AN ALGORITHM TO GENERATE CONNECTED GRAPHS* John Skvoretz University of South Carolina ABSTRACT Connected graphs hold special interest for network exchange researchers because all experimentally-studied networks are connected graphs. Competing theoretical proposals differ in the predictions they make for some sets of connected graphs, but not for other sets. Yet researchers do not have at their disposal any catalogue of connected graphs for even relatively small-sized groups. In this article, I review the theoretical basis for an algorithm that generates all connected graphs of a particular size and provide a pseudo-code version of the algorithm itself. I then address some considerations regarding its use in the network exchange field. [page 43] - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - [page 44] INTRODUCTION All of the exchange networks studied experimentally in the field of network exchange research are connected graphs, wherein the nodes are positions and the lines are exchange relations (i.e., relations in which agreement on the terms of exchange produces mutual benefit). Further, all positions are connected to one another either directly or indirectly. This connectedness of the set of positions is essential since the very nature of network exchange research focuses on how indirect connections affect the relative power of actors who are directly connected to one another. The graphs vary in size, with six positions being the largest for which published data exist (Skvoretz and Willer 1993). Network exchange research presently harbors several contending theories of power distribution. Evaluating these theories involves many considerations, including parsimony in the assumptive base, breadth of coverage, predictive precision, and embeddedness in accepted explanatory frameworks (e.g., rational choice theory). One important consideration is empirical accuracy: does one proposal make more accurate predictions than the others? For many graphs, the contenders make very similar predictions and so they cannot be used to assess accuracy. Rather, research must use those graphs (if any) for which the alternatives make different predictions. Currently, discovery of such candidates is a hit-or-miss proposition as there exists no accessible catalogue of connected graphs with which to systematically search for "critical" networks. In addition to the evaluation of empirical accuracy, a catalogue of connected graphs could aid in the development of particular approaches, as can be exemplified by Lovaglia, Skvoretz, Markovsky, and Willer (1995). These authors use a series of "thought experiments" to refine the Graph Power Index (GPI) approach first proposed by Markovsky, Willer, and Patton (1988): Refinements are proposed to solve problematic networks and then evaluated by inspecting the predictions offered for other networks of similar size or configuration. Only those refinements that offer sensible predictions are retained for empirically-based evaluation. Hence, a catalogue of connected graphs is needed grist for the "thought experiment" mill. [page 44] - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - [page 45] ALGORITHMS FOR GENERATING GRAPHS The problem of generating all connected graphs for a set p of points is a combinatorial problem. One procedure for generating combinatorial configurations of a certain type is called the "classical method" by Read (1978). In this procedure, there are two lists: one of old configurations and one of new configurations. Potential candidates for the new list are created by performing an operation on an old configuration, e.g., adding a node and an edge to one of the points in the old configuration. The candidate is added to the new list only if it is not already present. Doing this, however, requires checking the candidate against each of the items already present on the new list, a time-consuming process insofar as there are usually many isomorphic versions of any item on the new list, and each of these versions must be detected and discarded. For instance, the three networks in Figure 1 are isomorphic (i.e., they exhibit the same structure). Therefore, on any list of connected graphs of size 4, only one of them can appear. As the size of the graph and the number of edges grow, the combinatorial possibilities explode. A - B B - C A - C \ / \ / \ / C A B | | | D D D Figure 1. Three isomorphic graphs. Read (1978) proposes a general algorithm that does not require looking back over the new list to detect and discard duplicates. In this procedure, as each candidate is produced it can be immediately tested for inclusion on the new list by simply inspecting the configuration itself. Hence, Read titles his paper "Every One a Winner" because the candidate is added to the new list only if it is a winner. A principle advantage of this algorithm is that the list of new elements can be stored externally since searches of it are never required. Read's "orderly" algorithm may be described as follows. We must first specify which of all possible isomorphic configurations will represent its class in the list of configurations. This particular configuration is called the "canonical configuration." The center graph in Figure 1 is the canonical configuration for its isomorphism class. Canonicity is defined by representing the upper right triangle of the network's adjacency matrix as a binary integer, concatenating the first row of this triangle to the second, both to the third, and so on. The canonical configuration for the class is the one that produces the largest binary integer or code. The codes for the networks in Figure 1 are 110101, 111100, and 110110, the second one being the largest. [page 45] - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - [page 46] Second, we must specify an ordering of the list of canonical configurations. The binary integers corresponding to each canonical configuration form a logical basis for ordering the list. In fact, we arrange the list in descending order by these integers. Finally, we must define an augmenting operation by which one element from an old list can produce an ordered sequence of potential elements on a new list in the order they would appear on the new list if they were found to be acceptable candidates. The operation we use takes the binary code of a canonical configuration of p nodes and q edges and produces potential candidates for the list of p nodes and q+1 edges as follows. The binary code of the input configuration is scanned from right to left to find the rightmost non-zero element. If the code is 111100, this is the fourth element. Then the candidate configurations are created by changing in turn each of the trailing 0's to 1, proceeding from left to right. Thus, from 111100 we obtain two candidates, 111110 and 111101. The algorithm involves processing an old list of canonical configurations, taking each element in turn, and creating potential candidates for the new list. For each candidate, the algorithm must discover if it is a canonical configuration by inspecting permutations of rows and columns in the underlying adjacency matrix for an equivalent configuration with a larger code value. If this value ever equals or exceeds the code value of the last configuration added to the new list, the candidate is not a canonical configuration and processing can move on to the next candidate. If the configuration is canonical and its code value is less than the new list minimum, it is a "winner," allowing it to be added directly to the new list and the new list minimum to be updated. The program that actually generates all connected graphs of a given size is presented in pseudo-code form in Table 1. Note that it generates all graphs whether connected or not and then retains only those that are connected. The algorithm is not effective if its operation is confined to just connected graphs. There are connected graphs with q+1 edges whose code number is greater than other connected graphs with q+1 edges but whose production from the ordered list of canonical connected graphs with q edges will occur after the production of those graphs with lower code numbers. Consequently, they will never be added to the new list. However, they will be produced before these other graphs if the augmentation operation applies over all graphs, connected or not. [page 46] - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - [page 47] input nodes initialize file "g00" with bit-string of 0's {length =[nodes(nodes-1)]/2} for edges = 0 to [nodes(nodes-1)]/2 - 1 open file "g{nodes}{edges}" for input as #1 open file "g{nodes}{edges+1}" for output as #2 open file "g{nodes}{edges+1}c" for output as #3 initialize current minimum code value while not end-of-file #1 read string from #1 find rightmost non-zero element of string for each trailing zero change to 1 and create new adjacency matrix for each permutation of rows and columns compare binary number of permuted adjacency matrix with current minimum and if greater, try next trailing zero compare binary number of permuted adjacency matrix with current maximum and if greater, save as new maximum set current minimum =3D maximum write maximum to #2 if connected, write maximum to #3 wend close #1, #2, #3 end Table 1: Pseudo-code for an orderly algorithm to generate connected graphs Even though the orderly algorithm is relatively more efficient than the "classical" approach, it still becomes time consuming as the number of nodes increases. On a 90mhz Pentium it will take about 2 hours to generate all the connected graphs over 7 nodes, where one finds 853 non-isomorphic connected graphs. Over 8 nodes, there are 11,118 connected graphs. Consequently, generating the catalogue of these graphs takes literally days of CPU time. Thus, it is unreasonable for each researcher to generate his or her own catalogue of connected graphs. Instead, the catalogue, once generated, should be made widely accessible. Consequently, I supply an Appendix which catalogues all connected graphs from size 2 to size 7 and can supply upon request the catalogue of size 8 connected graphs. A Sun SparcStation is currently at work (on a part-time basis since December) generating all size 9 connected graphs. [page 47] - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - [page 48] CONCLUSION In addition to advancing theoretical inquiry, the catalogue of connected graphs can standardize nomenclature for experimental networks. For instance, the graph in Figure 2 is called "KITE" in the work of Markovsky, Skvoretz, Willer and colleagues (Markovsky, Skvoretz, Willer, Lovaglia, and Erger 1993; Skvoretz and Willer 1993) and "HOURGLASS" in the work of Bienenstock and Bonacich (1993). The catalogue provides a canonical name for this graph: namely, the number of nodes, followed by the number of edges, followed by its location in the canonical ordering. Thus the graph in Figure 2 has the canonical name "p5e6.2" because it is the second graph in the canonical listing of all connected graphs with 5 nodes or points and 6 edges. B - C \ / A / \ D - E Figure 2: The connected graph p5e6.2 The catalogue also provides a canonical way of labeling the nodes of an entry. One simply assigns letters to nodes in alphabetical order and then builds the adjacency matrix as specified by the bit string, thus standardizing how graphs are drawn for research reports. Both uses of the catalogue become more salient as the size of the networks investigated becomes larger. Investigators from different approaches can more easily communicate with one another over problematic graphs since, in switching to a standardized naming system, we avoid "pet" names that reflect accidental circumstances surrounding an approach's discovery. Thus, the actual use of the catalogue appears to depend on (a) maintaining theoretical heterogeneity in exchange network research, (b) extending empirical and theoretical investigation to relatively large networks (say, 7, 8 and 9 node networks) and (c) making accessible an organized catalogue of connected graphs. The track record of the network exchange research field over the last 10 years suggests condition (a) will hold for the foreseeable future. The present paper guarantees the satisfaction of condition (c). And the internal dynamic of theoretical development via conjecture and refutation pushes the explanatory envelope in the direction consistent with condition (b). Only empirical extension may lag for the usual budgetary reasons. [page 48] - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - [page 49] ENDNOTE *Reviewers provided helpful comments and suggestions. Address correspondence to: skvoretz-john@sc.edu REFERENCES Bienenstock, E.J. and P. Bonacich. 1993. "Game-Theory Models for Exchange Networks: Experimental Results." Sociological Perspectives 36: 117-35. Lovaglia, M., J. Skvoretz, B. Markovsky, and D. Willer. 1995. "Assessing Fundamental Power Differences in Exchange Networks: Iterative GPI." Current Research in Social Psychology 1(2): 8-15, http://www.uiowa.edu/~grpproc. Markovsky, B., J. Skvoretz, D. Willer, M. Lovaglia and J. Erger. 1993. "The Seeds of Weak Power: An Extension of Network Exchange Theory." American Sociological Review 58: 197-209. Markovsky, B., D. Willer, and T. Patton. 1988. "Power Relations in Exchange Networks." American Sociological Review 53: 220-36. Read, R.C. 1978. "Every One a Winner or How to Avoid Isomorphism Search When Cataloguing Combinatorial Configurations." Annals of Discrete Mathematics 2: 107-120. Skvoretz, J. and D. Willer. 1993. "Exclusion and Power: A Test of Four Theories of Power in Exchange Networks." American Sociological Review 58: 801-18. AUTHOR BIOGRAPHY John Skvoretz is a Carolina Distinguished Professor of Sociology at the University of South Carolina. Current research projects concern formal models of social networks, status and participation in discussion groups, power in exchange networks, and theoretical studies of action structures. He is the editor of Connections, the bulletin of the International Network for Social Network Analysis and an Associate Editor of the Journal of Mathematical Sociology.