Monotone graph limits and quasimonotone graphs
The recent theory of graph limits gives a powerful framework for understanding the properties of suitable (convergent) sequences $(G_n)$ of graphs in terms of a limiting object which may be represented by a symmetric function $W$ on $[0,1]$, i.e., a kernel or graphon. In this context it is natural t...
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| Format: | Journal article |
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2011
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| _version_ | 1826304926169432064 |
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| author | Bollobas, B Janson, S Riordan, O |
| author_facet | Bollobas, B Janson, S Riordan, O |
| author_sort | Bollobas, B |
| collection | OXFORD |
| description | The recent theory of graph limits gives a powerful framework for understanding the properties of suitable (convergent) sequences $(G_n)$ of graphs in terms of a limiting object which may be represented by a symmetric function $W$ on $[0,1]$, i.e., a kernel or graphon. In this context it is natural to wish to relate specific properties of the sequence to specific properties of the kernel. Here we show that the kernel is monotone (i.e., increasing in both variables) if and only if the sequence satisfies a `quasi-monotonicity' property defined by a certain functional tending to zero. As a tool we prove an inequality relating the cut and $L^1$ norms of kernels of the form $W_1-W_2$ with $W_1$ and $W_2$ monotone that may be of interest in its own right; no such inequality holds for general kernels. |
| first_indexed | 2024-03-07T06:25:07Z |
| format | Journal article |
| id | oxford-uuid:f403f1e7-390d-4d47-9812-aa7b57c66f22 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T06:25:07Z |
| publishDate | 2011 |
| record_format | dspace |
| spelling | oxford-uuid:f403f1e7-390d-4d47-9812-aa7b57c66f222022-03-27T12:16:35ZMonotone graph limits and quasimonotone graphsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:f403f1e7-390d-4d47-9812-aa7b57c66f22Symplectic Elements at Oxford2011Bollobas, BJanson, SRiordan, OThe recent theory of graph limits gives a powerful framework for understanding the properties of suitable (convergent) sequences $(G_n)$ of graphs in terms of a limiting object which may be represented by a symmetric function $W$ on $[0,1]$, i.e., a kernel or graphon. In this context it is natural to wish to relate specific properties of the sequence to specific properties of the kernel. Here we show that the kernel is monotone (i.e., increasing in both variables) if and only if the sequence satisfies a `quasi-monotonicity' property defined by a certain functional tending to zero. As a tool we prove an inequality relating the cut and $L^1$ norms of kernels of the form $W_1-W_2$ with $W_1$ and $W_2$ monotone that may be of interest in its own right; no such inequality holds for general kernels. |
| spellingShingle | Bollobas, B Janson, S Riordan, O Monotone graph limits and quasimonotone graphs |
| title | Monotone graph limits and quasimonotone graphs |
| title_full | Monotone graph limits and quasimonotone graphs |
| title_fullStr | Monotone graph limits and quasimonotone graphs |
| title_full_unstemmed | Monotone graph limits and quasimonotone graphs |
| title_short | Monotone graph limits and quasimonotone graphs |
| title_sort | monotone graph limits and quasimonotone graphs |
| work_keys_str_mv | AT bollobasb monotonegraphlimitsandquasimonotonegraphs AT jansons monotonegraphlimitsandquasimonotonegraphs AT riordano monotonegraphlimitsandquasimonotonegraphs |