Network Models: Growth, Dynamics, and Failure Sandip Roy Chalee Asavathiratham Bernard C. Lesieutre George C. Verghese Massachusetts Institute of Technology Abstract — This paper reports on preliminary explorations, both empirical and analytical, of probabilistic models of large-scale networks. We first examine the structure of net- works that grow by the addition of nodes and lines, using a class of connection rules motivated by considerations of dis- tance and prior connectivity. Second, we examine the dy- namic behavior of the binary influence model — a particular form of a more general model of networks in which each node has a status (for instance: normal, or failed) that behaves as a Markov chain, but with transitions that are influenced by the present status of each neighboring node. Some in- teresting influence model examples are analyzed, including one displaying a power-law relation between the frequency of a failure event and its extensiveness. I. Introduction Extensive failures in large complex systems often result fromcascadingevents,inwhichaninitialfailureofonenet- work component impacts other components and induces their failure, eventually impacting a large portion of the system. The tendency for cascading failures in designed networks is not completely understood, but is necessarily influenced by the underlying structure of these networks. Forexample,asystemcomprisingacollectionofsmalldis- connected regions will not exhibit large-scalefailures. The interconnection of a wide area introduces the possibility for large-scale failures. In addition, the dynamic behavior of network components is important in explaining the oc- currence of cascading failures. The dynamics of designed networksarevariedandareoftencomplex,sothatthepos- sibility for cascading failures in these networks is difficult to predict in general. Understanding the nature of inter- connectionsanddynamicsthatwillmakeextensivefailures probableisoneofthegoalsofthisresearch. Several approaches for understanding failures in net- works have been discussed in the literature. From real data,researchershavetabulatedstatisticsofspecificfailure events in certain types of networks. One such representa- tion, for the electric power grid, is shown in Figure 1 (the data are from [1]). The statistics represented in this plot providecompellingevidencethatlarge-scaleoutagesonthe power grid are not anomalies, but should be expected as infrequent, but regular, occurrences. While such a high- levelstatisticalrepresentationofdataisuseful,itdoesnot explain why such behavior is observed, nor does it help identify which features of the network structure and dy- namicsmostinfluenceit,orsuggesthowtoapplyresources to mitigate failures. A complementary approach is the study of failures us- ing appropriate models of networks. The models can be usedinatleasttwodifferentways. First,onemayperform Fig. 1. A plot of cumulative frequency versus number of customers affected by power system disturbances, 1984-1997. numerous studies to obtain statistical data similar to that compiled in the real world observations. (And if the mod- els adequately represent real-world systems, the resulting statisticsshouldbesimilar.) Correlatingthestatisticswith certainquantitiesrepresentedinthemodelmayyieldmore informationaboutthesystem. Second,onemayattemptto analyticallyrelatefeaturesofthenetworkanddynamicsto resulting system behavior. While very difficult to obtain, such relations would be the most valuable results. Inthisarticlewelookatgrowthmodelsofnetworksus- ing simple connection rules. We study the connectivity propertiesoftheresultingnetworks,someofwhichdisplay power-law characteristics. Later in the paper we compare the connectivity statistics to failure statistics on the same networks. For dynamic studies we introduce the Influence Model, a general probabilistic model to describe both the occur- renceofinitialfailureeventsinnetworksandtheinfluence ofthesefailuresonotherpartsofthenetwork. Intheinflu- ence model each node has a status (for instance: normal, or failed) that behaves as a Markov chain, but the tran- sitions of each chain are influenced by the present status ofeachneighboringnode. Theoverallnetworkcanstillbe described as a Markov chain, but with order equal to the product of the orders of the individual node chains. We have established that analysis of a greatly reduced model,